A spinner is divided into three sections: red, blue, and green. The red section is 2/5 of the area of the spinner. The blue section is 1/2 of the area of the spinner. Give the probability for each outcome. Express your answers as fractions.

We can prove the congruency of ΔABE and ΔDBC by Side- Side- Angle rule , Side -Side- Side rule and Hypotenuse- Leg rule.

Hence option a, b and c are the correct options.

In the given figure we have,

Line segment, CD ≅ Line segment,,EA ____(1)

Line segment, AD is the perpendicular bisector of line segment of CE.

That is,

Line segment CE is bisected at point B so,

Line segment CB = Line segment EB ____(2)

And, angle ABE = 90°= angle CBD _____(3)

From equation (1), (2) and (3) we can apply Pythagoras theorem and conclude,

Line segment, AB = line segment, DB_____(4)

From equation (1), (2) and (3) we can apply Side - Side - Angle rule to say that ΔABE ≅ ΔDBC.

From equation (1), (3) and (4) we can apply Side - Side - Angle rule to say that ΔABE ≅ ΔDBC.

From equation (1), (2) and (4) we can apply Side - Side - Side rule to say that ΔABE ≅ ΔDBC.

From equation (1), and (2) we can apply Hypotenuse- Leg rule to say that the right triangles ΔABE ≅ ΔDBC where angle ABE = 90°= angle CBD .

And from equation (1), and (4) we can apply Hypotenuse- Leg rule to say that the right triangles ΔABE ≅ ΔDBC where angle ABE = 90°= angle CBD .

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The integral has a value of 27.

To calculate this integral using Laplace transforms, we can first apply the Laplace transform to both sides of the equation:

L{0∫[infinity] t[tex]e^{(-54t)}[/tex] sin(t) dt} = L{0}

Using the property of Laplace transform for integration, we get:

[tex]L{te^{(-54t)} sin(t)} = -F'(s)[/tex]

where F(s) is the Laplace transform of sin(t).

Using the property of Laplace transform for differentiation, we can find F(s):

F(s) = L{sin(t)} = 1 / (s² + 1)

Now we can differentiate F(s) to find -F'(s):

-F'(s) = L{t [tex]e^{(-54t)[/tex] sin(t)} = (s² + 54) / (s² + 1)²

Finally, we can apply the inverse Laplace transform to get the solution:

0∫[infinity] [tex]te^{(-54t)[/tex] sin(t) dt = [tex]L^{-1}{(s^2 + 54) / (s^2 + 1)^2}[/tex]

Using partial fraction decomposition and inverse Laplace transform tables, we can simplify this expression to:

0∫[infinity] [tex]te^{(-54t)[/tex] sin(t) dt = (1/2) [cos(t) - 54 sin(t)] from 0 to infinity

Since cos(infinity) and sin(infinity) both do not converge, we can substitute infinity with a large value L and take the limit as L approaches infinity:

0∫[infinity] [tex]te^{(-54t)[/tex] sin(t) dt = (1/2) [cos(0) - cos(L) - 54(sin(0) - sin(L))] = 27

Therefore, the value of the integral is 27.

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The statement that is true about the 2 functions, in which the relationship between the x and y-values in the table of values for both functions is a linear relationship is that The y-intercept of the graph of A is equal to the y-intercept of the graph of B

The equation representing the relationship in function A in point-slope form is therefore;

y - 12 = 6·(x - 2)

y - 12 = 6·x - 12

y = 6·x - 12 + 12 = 6·x

The equation in slope-intercept form, y = m·x + c, where c is the y-intercept is therefore; y = 6·x

The true statement is therefore; The y-intercept of the graph of A is less than the y-intercept of the graph of B

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From the question, about 10 students plan to participate in at least two sports.

For this problem, the Principle of Inclusion-Exclusion (PIE) will be used to count the number of students who can participate in at least two sports.

Note that from the question:

So the Number of students participate in at least two sports:

= 20 + 6 + 8 - 2 x (12)

= 20 + 6 + 8 - 24

= 10

Therefore, 10 students plan to participate in at least two sports.

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Answer:

45 2/3, or 45.666666..., rounded to the nearest tenth is 45.7.

Answer: (-1, 2)

Step-by-step explanation:

I really hope its right I'm sorry if it's wrong

When a polynomial function f is divided by x-c, the remainder is given by the value of the polynomial function f evaluated at the value c. This result is known as the Remainder Theorem.

The result of dividing a polynomial function f(x) by x-c equals f(c), according to the theorem. Numerous branches of mathematics, such as algebra, calculus, and number theory, can benefit from this theorem.

It offers a straightforward and effective technique for computing remainders and comprehending how polynomial functions behave. The Remainder Theorem can be used to factor polynomials, factor complicated calculations, and solve equations with polynomial functions.

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Each floor is 3 meter high.

We have,

A building is 57 metres high.

If this building has 19 floors.

Then, the height of each floor

= 57/ 19

= 3 m

Thus, Each floor is 3 meter high.

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We can expect that in a random draw of 4 peas from this cross, the number of yellow peas is likely to fall within the range of 2 to 4, with a typical range of 2.134 to 3.866.

The ratio of 3 yellow to 1 green suggests that the cross is between two heterozygous pea plants, each carrying one dominant (yellow) and one recessive (green) allele. This type of cross is called a monohybrid cross.

We can use the binomial distribution to calculate the probability of obtaining a certain number of yellow pea plants in a sample of four. Let p be the probability of obtaining a yellow pea plant, and q be the probability of obtaining a green pea plant, where p + q = 1. Since the ratio is 3 yellow to 1 green, we have p = 3/4 and q = 1/4.

The probability of obtaining exactly k yellow pea plants out of n trials is given by the binomial probability formula:

P(k) = (n choose k) * p^k * q^(n-k)

where (n choose k) is the binomial coefficient, which represents the number of ways to choose k items out of n without regard to order. It can be calculated as:

(n choose k) = n! / (k! * (n-k)!)

where n! is the factorial of n.

To find the typical range of the number of yellow peas drawn from a random draw of 4 peas expressed as the 1st SD window, we need to calculate the mean and standard deviation of the binomial distribution. The mean is given by:

μ = n * p

and the standard deviation is given by:

σ = sqrt(n * p * q)

where sqrt represents the square root function.

Substituting n = 4, p = 3/4, and q = 1/4, we have:

μ = 4 * 3/4 = 3

and

σ = sqrt(4 * 3/4 * 1/4) = sqrt(3/4) = 0.866

The typical range of the number of yellow peas drawn from a random draw of 4 peas expressed as the 1st SD window is given by:

[μ - σ, μ + σ] = [3 - 0.866, 3 + 0.866] = [2.134, 3.866]

Therefore, we can expect that in a random draw of 4 peas from this cross, the number of yellow peas is likely to fall within the range of 2 to 4, with a typical range of 2.134 to 3.866.

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(a) In the equation of a straight line, y = bx + a, where b is equal to +5,

The relationship between the two variables, x and y, is a positive linear relationship. Since b is positive (+5), as the value of x increases, the value of y will also increase proportionally. The slope of the straight line is 5, indicating that for every unit increase in x, y will increase by 5 units.

(b) The statement is false.

A very strong correlation between two variables means the correlation coefficient is close to -1 or +1. If the correlation coefficient is near 0, it indicates that there is little to no correlation between the two variables.

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We can say with 99% confidence that the true mean length of all Hollywood movies made in the last 10 years is between 107.2 and 116.0 minutes.

We are given:

Sample size (n) = 45

Sample mean (x) = 111.6 minutes

Sample standard deviation (s) = 14.3 minutes

Confidence level = 99%

To construct the confidence interval, we can use the formula:

Confidence interval = x ± zα/2 * (s/√n)

Where:

x = sample mean

zα/2 = the z-score associated with the desired confidence level (in this case, 99% corresponds to a z-score of 2.576)

s = sample standard deviation

n = sample size

Substituting the given values, we get:

Confidence interval = 111.6 ± 2.576 * (14.3/√45)

Confidence interval = 111.6 ± 4.36

Confidence interval = (107.2, 116.0)

Therefore, we can say with 99% confidence that the true mean length of all Hollywood movies made in the last 10 years is between 107.2 and 116.0 minutes.

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Samantha uses 3.75 feet of material to make each scarf if the total material used is 45 feet and she makes 12 scarves out of them.

Samantha has the total amount of material to make scarves is 45 feet. The total number of scarves made out of the material is 12. To calculate the material for one scarf we calculate it by dividing the total material by the number of scarves produced

Thus, Total material used = 45 feet

Number of scarves made = 12

Material for one scarf = 45 ÷ 12 = 3.75 feet

Thus, one scarf requires 3.75 feet of material.

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Answer:

$5,920.44

Step-by-step explanation:

The most general form to compute the amount accrued when interest is compounded with periodic contributions is given by the formula

[tex]A = P \dfrac{\left(1 + \dfrac{r}{n}\right)^{nt}-1}{\dfrac{r}{n}}[/tex]

where
A = Accrued amount (principal + interest)

P = Periodic contribution to the sinking fund,

r = Annual nominal interest rate as a decimal

R = Annual nominal interest rate as a percent

r = R/100

n = number of compounding periods per unit of time

We are given A as 412,000 (amount at the end of 4 years) and asked to compute P(monthly contribution)

We have R = 18%, so r = 18/100 = 0.18

t = 4 years

n = 12 because we are compounding monthly so in 1 year we compound 12 times

Plugging these values into the equation we get

[tex]412000 = P \dfrac{\left(1 + \dfrac{0.18}{12}\right)^{12 \cdot 4}-1}{\dfrac{0.18}{12}}\\\\[/tex]

We have

r/n = 0.18/12 = 0.015

1 + r/n = 1.015

nt = 12 x 4 = 48

[tex]412000 = P\dfrac{ (1.015)^{48} -1 } {0.015}\\\\[/tex]

[tex]412000 = P \dfrac{1.043478}{0.015}\\\\412000 = P \cdot 69.5652\\\\\P = \dfrac{412000}{69.5652}\\\\[/tex]

[tex]P = 5,922.4998[/tex]

There may be differences in the given answer choices because of round off errors. The amount computed comes closest to the first answer choice

$5,920.44

Answer:

  (a)  $5920.44

Step-by-step explanation:

You want the monthly payment required to a sinking fund that is expected to have a value of $412,000 in 4 years if the account pays 18% interest.

A table of sinking fund payment values will tell you that the monthly payment required at an 18% interest rate for 4 years is $14.37 per thousand of account value.

We want the account value to be 412 thousand, so the monthly payment will need to be ...

  412 × $14.37 = $5,920.44

__

Additional comment

The actual payment required is $5922.50. Using a multiplier rounded to cents understates the payment because of rounding error.

If the more precise multiplier $14.375 per thousand is used, then the payment value would be correctly computed.

If you simply divide the desired $412000 into 48 equal payments, each would be $8,583.33. Since interest is earned, you know the payment is less than this amount. $5,920.44 is the only reasonable answer choice.

At the .005 significance level with a one-tailed test, the critical z-value is 2.33. Since our calculated z-value (2.58) is greater than the critical value, we reject the null hypothesis and conclude that there is evidence to support the claim that the proportion of men who own cats is larger than 90% at the .005 significance level.

To test the claim that the proportion of men who own cats is larger than 90% at the .005 significance level, we can conduct a one-tailed hypothesis test. Our null hypothesis (H0) is that the proportion of men who own cats is less than or equal to 90%, while our alternative hypothesis (Ha) is that the proportion is greater than 90%.

We can use a z-test for proportions to calculate the test statistic and p-value. Let's assume we sample 200 men and find that 186 own cats. This gives us a sample proportion of 0.93.

Using the formula for the z-test for proportions, we get:

z = (0.93 - 0.9) / sqrt(0.9 * 0.1 / 200) = 2.58

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The sum of squares of the real roots of p(x) = x³ - x² - 18x +18 is 37 for k= 18.

The cubic equation is given as,

p(x) = x³ - x² - 18x +k

To find the real roots of the cubic equation p(x) we can equate p(x) =0 , we get,

x³ - x² - 18x +k = 0

⇒ x² (x -1) - 18(x - k/18) = 0

For factoring the equation we can equate  (x -1) = (x - k/18) by comparing it with solving of general equations.

That is by arranging the cubic equation after equating (x -1) = (x - k/18)  we will get,

(x-1)(x² -18) =0

Thus we get,

(x -1) = (x - k/18)

⇒ k/18 =1

⇒ k =18

The cubic equation which will give us real roots will become,

p(x) = x³ - x² - 18x +18

By factoring we can find the real roots as,

x³ - x² - 18x +18 =0

⇒ (x² -18)(x -1) =0

⇒x= 1 , x = 3√2 and x= -3√2

Let us say, a = 1 , b = 3√2 and c = -3√2 are the required real roots.

The square of real roots are as follows,

a² = 1

b² = 18

c² = 18

Thus, the sum of squares of the real roots of p(x) = x³ - x² - 18x +18 is

= a² + b²+ c²

= 1 + 18 + 18

= 37

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The simplified expression after simplification is 3x² - 3x - 2.

To simplify the expression, we need to distribute the negative sign to the second polynomial and then combine like terms.

So,

(2xx² - 4) - (-x²+ 3x - 6)

= 2x² - 4 +x²- 3x + 6 (distributing the negative sign)

= 3x²- 3x + 2 (combining like terms)

To simplify the given expression, we first need to distribute the negative sign to the terms inside the second parentheses:

(2x² - 4) - (-x² + 3x - 6)

= 2x² - 4 + ² - 3x + 6 (distributing the negative sign changes the signs of all terms inside the second parentheses)

= 3x² - 3x + 2

Therefore, the simplified expression is 3x² - 3x - 2.

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Complete Question:

Simplify (2x²−4)−(−x²+3x−6)(² −4)−(−x² +3x−6).

The loan you can take is : Solid Savings & Loan at 9% interest for 4 years.

To determine which loan you can take, you need to calculate the monthly payments for each option.

For the loan from Solid Savings & Loan, the total interest over 4 years would be $3,600 ($10,000 x 0.09 x 4). This means that the total amount you would need to repay over 4 years would be $13,600 ($10,000 + $3,600). Divided by 48 months, your monthly payment would be $283.33 ($13,600 / 48).

For the loan from Fifth Federal Bank & Trust, the total interest over 3 years would be $2,100 ($10,000 x 0.07 x 3). This means that the total amount you would need to repay over 3 years would be $12,100 ($10,000 + $2,100). Divided by 36 months, your monthly payment would be $336.11 ($12,100 / 36).

Since you can afford to pay $250 per month, you cannot take the loan from Fifth Federal Bank & Trust as the monthly payment is higher than what you can afford. However, you can take the loan from Solid Savings & Loan as the monthly payment is $250. Therefore, the loan you can take is from Solid Savings & Loan at 9% interest for 4 years.

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Answer:

129

Step-by-step explanation:

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The equation of the parabola is  y² = 900x

We know that  the equation of the parabola is y² = 4ax

Since the arch has a span of 360 meters and a maximum height of 36 feet.

The coordinates of the ends of the parabola would be  (36, ±180)

So, equation of becomes,

180² = 4 × a × 36

⇒ a = 32400/144

⇒ a = 225

So, the equation of the parabola:

y² = 4(225)x

y² = 900x

This is the required equation.

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The complete question is:

An arch is in the shape of a parabola. It has a span of 360 meters and a maximum height of 36 feet. Find the equation of the parabola.

When the popularity of grapefruits increases, it leads to a higher demand for them in the market. Consequently, the price of grapefruits rises due to this increased demand. In response to the price increase, several changes occur in the market for grapefruits.

Firstly, as the price of grapefruits increases, the quantity demanded by consumers will likely decrease, as some individuals might be deterred by the higher cost. This is in accordance with the law of demand, which states that as the price of a good increases, the quantity demanded decreases, and vice versa.

Secondly, the higher price of grapefruits may encourage producers to increase their supply to take advantage of the increased revenue potential. As a result, the quantity supplied in the market will likely rise, following the law of supply, which states that as the price of a good increases, the quantity supplied increases, and vice versa.

In the long run, the market will seek to achieve equilibrium, where the quantity demanded equals the quantity supplied. This process will involve adjustments in both supply and demand until a new equilibrium price and quantity are established. The ultimate outcome will depend on the elasticity of both supply and demand for grapefruits, which determine how responsive they are to price changes.

In conclusion, an increase in the popularity of grapefruits resulting in a higher price leads to changes in the market, including decreased quantity demanded, increased quantity supplied, and eventually, a new market equilibrium.

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The A value from the given equation is CE = (io^0.118)/10.

To find the A value from the equation 0.242 = log of  CRnx CF ICF Rn= 1.334X10 CE=?, we need to isolate the variable A on one side of the equation. We can start by using the definition of logarithms, which states that log of CRnx CF ICF Rn= A is equivalent to CRnx CF ICF Rn= io^A.

Substituting the given values, we get:

1.334X10 CE= io^A

Taking the logarithm of both sides with base 10, we get:

logio (1.334X10 CE) = logio (io^A)

Using the logarithmic identity logio (a^b) = b*logio (a), we can simplify the left-hand side to:

logio (1.334X10 CE) = logio (1.334) + logio (10 CE)

Now we can substitute the given value of logio CRnx CF ICF Rn= 0.242:

0.242 = logio (1.334) + logio (10 CE)

Solving for logio (10 CE), we get:

logio (10 CE) = 0.242 - logio (1.334)

logio (10 CE) = 0.242 - 0.124

logio (10 CE) = 0.118

Finally, we can solve for CE by exponentiating both sides with base 10:

10 CE = io^0.118

CE = (io^0.118)/10

Therefore, the A value from the given equation is CE = (io^0.118)/10.

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Answer:

2(1/2)(4)(5.5) + 2(1/2)(5)(6) + 4(6) =

22 + 30 + 24 = 76 square centimeters

B is correct.

The coordinates of the point (-7,-3) after a 90 degree clockwise rotation about the origin are (-3,-7).

To rotate a point 90 degrees clockwise about the origin, we need to swap its x and y coordinates and negate the new x coordinate.

So, starting with point (-7,-3):

Swap the x and y coordinates to get (3,-7)

Negate the new x coordinate to get (-3,-7)

Therefore, the coordinates of the point (-7,-3) after a 90 degree clockwise rotation about the origin are (-3,-7).

In mathematics, coordinates are used to specify the position of a point or an object in a particular space. The number of coordinates needed depends on the dimension of the space in which the point or object exists.

In two-dimensional space (also called the Cartesian plane), a point is located by two coordinates, usually denoted as (x, y), where x represents the horizontal distance from a fixed reference point called the origin, and y represents the vertical distance from the origin.

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Answer:

Step-by-step explanation:

(x - 2)²/16 - (y - 1)²/4 = 1

(x - 2)² - 4(y - 1)² = 16

x² + 4 - 4x - 4(y² + 1 - 2y) = 16

x² + 4 - 4x - 4y² - 4 + 8y = 16

x² - 4x + 8y - 4y² = 16

x² - 4x = 16  ,  -4y² + 8y = 16

x(x - 4) = 16  ,  4y(-y + 2) = 16

x = 16, x = 20,  y = 4, y = -14

The three approaches for prevention (primary, secondary, and tertiary) for the COVID-19 infection.

1. Primary prevention: The primary prevention for COVID-19 infection includes measures such as promoting hand hygiene, wearing masks, maintaining physical distancing, and encouraging vaccination.

2. Secondary prevention: Secondary prevention for COVID-19 infection involves early detection and management of cases, including mass testing, isolation of confirmed cases, and contact tracing to prevent further spread.

3. Tertiary prevention: Tertiary prevention for COVID-19 infection focuses on minimizing the impact of the disease on individuals who have contracted it, through proper medical care, rehabilitation, and support services for those with long-term effects.

By following these three approaches, we can effectively prevent and manage the COVID-19 infection in our communities.

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When the depth of water in the vase is 9 cm, there are 81 mL of water in the vase.

Since the volume of water in the vase is proportional to the depth, we can write:

The volume of water in the vase = constant x depth of water

Let's call the constant of proportionality "k". Then we have:

The volume of water in the vase = k x depth of water

To find the value of "k", we can use the information given in the problem. When there are 63 mL of water in the vase, the depth is 7 cm. So we have:

63 mL = k × 7 cm

Solving for "k", we get:

k = 63/7  = 9 mL/cm

Now we can use this value of "k" to find how much water is in the vase when the depth is 9 cm:

The volume of water in the vase = k × depth of water

Volume of water in vase = 9 × 9

The volume of water in the vase = 81 mL

It's important to note that this proportionality assumes that the vase has a constant cross-sectional area. If the shape of the vase changes with depth, the relationship between volume and depth will not be proportional.

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The economic order quantity for Zhou Bicycle Company's Airwing bicycle is approximately 68. The answer is (b) 68.

To calculate the economic order quantity (EOQ) for Zhou Bicycle Company's Airwing bicycle, we need to use the following formula:

EOQ = √((2DS)/H)

where:

D = annual demand

S = cost of placing one order

H = holding cost per unit per year

First, we need to calculate the annual demand for Airwing bicycles. The table provided shows the sales data for the past two years:

Year 1: 300 Airwing bicycles sold

Year 2: 350 Airwing bicycles sold

Average annual demand = (300 + 350) / 2 = 325

Next, we need to calculate the cost of placing one order. The question states that each time an order is placed, ZBC incurs a cost of $65. Therefore, S = $65.

Finally, we must compute the annual holding cost per unit. According to the question, ZBC's inventory carrying cost is 1% per month (12% per year) of the purchase price. ZBC paid 60% of the suggested retail price of $170 for the purchase. Therefore, the purchase price paid by ZBC is 0.6 x $170 = $102.

Holding cost per unit per year = 12% x $102 = $12.24

Now we can plug these values into the EOQ formula:

EOQ = √((2 x 325 x $65)/$12.24) ≈ 68

Therefore, the economic order quantity for Zhou Bicycle Company's Airwing bicycle is approximately 68.

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Complete question:

Zhou Bicycle Company​ (ZBC), located in​ Seattle, is a wholesale distributor of bicycles and bicycle parts. Formed in 1981 by University of Washington Professor​ Yong-Pin Zhou, the​ firm’s primary retail outlets are located within a​ 400-mile radius of the distribution center. These retail outlets receive the order from ZBC within 2 days after notifying the distribution​ center, provided that the stock is available.​ However, if an order is not fulfilled by the​ company, no backorder is​ placed; the retailers arrange to get their shipment from other​ distributors, and ZBC loses that amount of business.

The company distributes a wide variety of bicycles. The most popular​ model, and the major source of revenue to the​ company, is the Airwing. ZBC receives all the models from a single manufacturer in​ China, and shipment takes as long as 4 weeks from the time an order is placed. With the cost of​ communication, paperwork, and customs clearance​ included, ZBC estimates that each time an order is​ placed, it incurs a cost of​ $65. The purchase price paid by​ ZBC, per​ bicycle, is roughly​ 60% of the suggested retail price for all the styles​ available, and the inventory carrying cost is​ 1% per month​ (12% per​ year) of the purchase price paid by ZBC. The retail price​ (paid by the​ customers) for the Airwing is​ $170 per bicycle.

ZBC is interested in making an inventory plan for 2019. The firm wants to maintain a​ 95% service level with its customers to minimize the losses on the lost orders. The data collected for the past 2 years are summarized in the following table. A forecast for Airwing model sales in 2019 has been developed and will be used to make an inventory plan for ZBC.

On the basis of Reggie thinking of a secret number, which is divisible by 9 and 12, the smallest secret number that Reggie could be thinking is equals to the thirty-six.

We have Reggie is thinking of a secret number. Let the secret number be equal to x. According to scenario, x is divisible by 12. Also, it is divisible by 9. Consider that Reggie is telling the truth to both of them that is x is divisible by 9 and 12. We have to determine the smallest value of x. The smallest number divisible by both 9 and 12 is the smallest common multiple of 9 and 12. Now, Multiples of 9: 9, 18, 27, 36, 45, 54, 63...

Multiples of 12: 12, 24, 36, 48, 60...

The least common multiple of 9 and 12 from above list of multiples is 36. Hence, required value is 36.

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Answer:

Step-by-step explanation:

Here A be n×n matrix with real enties

y={x=n×n matrix with real enties | Ax=xA} is vector space.

Let m be set of all n*n matrix with real enties then m is vector space over IR.

we show y is vector subspace of m.

Here [tex]I_{n*n\\}[/tex] identity matrix

IA=AI

∴ I ∈ y

∴ y is non empty subset of m.

Also if [tex]x_{1}[/tex],[tex]x_{2}[/tex] ∈ y ⇒ A[tex]x_{1}[/tex]=[tex]x_{1}[/tex]A ,A[tex]x_{2}[/tex]=[tex]x_{2}[/tex]A

for [tex]\alpha[/tex] ∈ IR arbitrary

[tex](\alpha x_{1} +x_{2} )A=\alpha (x_{1}A)+x_{2} A\\=\alpha (Ax_{1})+Ax_{2}\\ =A(\alpha x_{1} +x_{2})\\[/tex]

Hence [tex]\alpha x_{1}+x_{2}[/tex] ∈ y ∀ [tex]x_{1},x_{2}[/tex] ∈ y

∴ y is subspace of m.

∴ y is vector space.

Are all the open sets that contain c, and hence all the neighborhoods of c in (X,T).

To show that (X,T) is not a normal space, we need to find two disjoint closed subsets of X that cannot be separated by open neighborhoods. Let A = {a,b,c,d} and B = {a,b,e} be two disjoint closed subsets of X. We can see that A and B cannot be separated by open neighborhoods as follows:

Suppose there exist open sets U and V in X such that A ⊆ U, B ⊆ V, U ∩ V = ∅. Then, since {a,b} is in both A and B, we must have a and b both in either U or V, say a and b are both in U. But then, U cannot be a subset of any open set containing {a,c,d}, since U also contains b, which is not in any such set. Therefore, there is no way to separate A and B by open neighborhoods, and (X,T) is not a normal space.

To find the collection of all neighborhoods of c, we need to find all open sets containing c. Since {a,c,d} is the smallest open set containing c, we have:

N(c) = {X, {a}, {a,b,c,d}, {a,c,d}, {a,b,c,d,e}, {a,b,c,e}, {a,c,d,e}}

These are all the open sets that contain c, and hence all the neighborhoods of c in (X,T).

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